Introduction
The concrete stair formula looks almost too simple: steps × width × tread × riser ÷ 2. Where does the ÷ 2 come from? Why is the step count rounded up instead of to the nearest whole number? And what does the triangle approximation leave out? This guide derives each piece from the geometry, so you can trust the number — and know exactly where its limits are.
The Concrete Stair Calculator runs this exact derivation automatically: enter your total rise, tread depth, and width, and it returns the volume in cubic meters and cubic yards plus a bag count. What follows is the math behind that result, worked in both metric and imperial.
Step 1: From Total Rise to Number of Risers
Everything starts with the total rise — the vertical distance between the two finished surfaces the stair connects. From it, the number of risers is derived, not chosen:
Number of risers = Total rise ÷ Target riser height, rounded UP
The rounding direction matters. Take a total rise of 120 cm with a target riser height of 18 cm (the typical code maximum): 120 ÷ 18 = 6.67. Rounding down to 6 gives risers of 20 cm — over the limit and illegal under most codes. Rounding up to 7 gives:
Actual riser height = Total rise ÷ Number of risers = 120 ÷ 7 = 17.14 cm (6.75 in)
Every riser in the flight is identical at 17.14 cm. Uniform risers aren’t just a code requirement — they’re a safety one. People calibrate their stride to the first step; a single odd riser is a trip hazard. The code-compliance guide covers the riser uniformity rules inspectors check.
Step 2: The Triangle at the Heart of the Formula
Look at one step from the side. Its profile is a rectangle (the tread surface and riser face) — but the concrete added by that step, the wedge of material that makes it a step rather than a ramp, is a right triangle with legs equal to the tread depth (t) and the riser height (r):
Area of one step’s profile = t × r ÷ 2
Extrude that triangle across the stair width (w) and you get one step’s volume: t × r ÷ 2 × w. Stack n identical steps and the flight’s geometric volume is:
Volume = n × w × t × r ÷ 2
That’s the whole formula. The ÷ 2 is the triangle area factor — each step contributes half of its tread-by-riser rectangle, because the other half is the “missing” wedge that makes it a step instead of a solid block.
What the triangle model leaves out
The triangle formula models steps formed directly on grade or on a sloped base — the common case for porch, patio, and landscape stairs. It does not include:
- The waist (throat) thickness of a structural stair with a sloped soffit — the solid slab beneath the steps. Its volume is waist thickness × slope length × width, where slope length = √(total run² + total rise²). Add it separately for elevated structural stairs.
- Landings — separate rectangular slabs: length × width × thickness.
- Footings and cheek walls — separate rectangular or triangular prisms.
For ordinary steps on grade, the triangle model is accurate to within the waste allowance. For structural stairs, add the waist — the estimating guide shows how.
Step 3: Total Run and the Top-Tread Question
The formula needs tread depth but not total run — still, you’ll need the run to lay out the formwork:
Total run = Number of treads × Tread depth
One subtlety: the top “tread” is often the porch or landing slab itself, not a formed step. With 7 risers and 28 cm treads, the flight spans 7 × 28 = 196 cm (77 in / 6.42 ft) if every tread is formed — but only 6 × 28 = 168 cm (66 in / 5.5 ft) of formed run if the seventh tread is the existing porch. The concrete volume is the same either way (the triangle math counts risers), but the formwork layout differs. Confirm which one your stair uses before cutting lumber.
Step 4: Waste Allowance
Geometric volume assumes perfect forms and zero spillage. Real pours get:
Order volume = Geometric volume × (1 + Waste%)
Ten percent is the standard default. Stairs deserve the full 10% — sometimes a little more — because stepped formwork has many joints to leak through and many small pockets where concrete hangs up. The waste table in the complete guide lists suggested allowances by situation.
Step 5: Unit Conversions and Bag Count
The volume conversions are the same as for any concrete estimate: 1 m³ = 1.308 yd³ (divide m³ by 0.7645549 for precision), 1 yd³ = 27 ft³. The bag count is:
Bags = Order volume ÷ Yield per bag, rounded UP
Standard approximate yields: 80-lb bag ≈ 0.6 ft³ (0.017 m³); 60-lb bag ≈ 0.45 ft³ (0.0127 m³); 40-lb bag ≈ 0.3 ft³ (0.0085 m³). Always round up — a half-bag shortfall at the top step can’t be fixed without a visible cold joint.
Full Worked Derivation: 1.2 m Porch Stair
Inputs: total rise 120 cm, tread 28 cm, width 1.2 m, max riser 18 cm, 10% waste.
Risers: 120 ÷ 18 = 6.67 → 7 risers; actual riser = 120 ÷ 7 = 17.14 cm.
Triangle area per step: 0.28 m × 0.1714 m ÷ 2 = 0.0240 m².
Geometric volume: 7 × 1.2 m × 0.0240 m² = 0.2016 m³.
With waste: 0.2016 × 1.10 = 0.2218 m³ → ÷ 0.7645549 = 0.29 yd³.
Bags: 0.2218 ÷ 0.017 = 13.05 → 14 bags (80-lb).
Imperial cross-check — 7 risers × 3.94 ft wide × 11 in tread × 6.75 in riser: triangle area per step = (11/12) × (6.75/12) ÷ 2 = 0.2578 ft²; × 3.94 ft × 7 = 7.11 ft³ = 0.263 yd³, matching the metric 0.2016 m³ = 0.264 yd³ (difference is rounding). With waste: 7.82 ft³ ÷ 0.6 = 13.03 → 14 bags. Both systems agree.
Related Guides
- Concrete Stair Complete Guide — the full estimating walkthrough.
- Estimating Guide — the measuring workflow that feeds this formula.
- Stair FAQ — quick answers on risers, treads, and volumes.
- Common Mistakes — where the math usually goes wrong in practice.
- Case Study — the formula applied to a hypothetical porch project.
- Pour-Day Checklist — verify the numbers before the concrete arrives.
Frequently Asked Questions
Why is stair volume divided by 2?
Each step’s concrete profile is a right triangle with legs equal to the tread depth and riser height. A triangle’s area is half its enclosing rectangle (t × r ÷ 2); extruded across the stair width, that gives each step’s volume.
How do I calculate the number of risers?
Divide the total rise by your target riser height (no more than the code maximum, typically 18 cm / 7 in) and round UP. Then actual riser height = total rise ÷ number of risers, keeping every riser identical and within limits.
Does the formula include the landing?
No. Estimate landings separately as rectangular slabs (length × width × thickness) and add them to the step volume. The same goes for footings, cheek walls, and any waist slab under a structural stair.
How accurate is the triangle approximation?
For steps formed on grade — porch, patio, and landscape stairs — it’s accurate to within the normal 10% waste allowance. For structural stairs with a sloped soffit, add the waist slab volume (waist thickness × slope length × width) separately.
Should the top tread count as a step?
For volume, it doesn’t matter — the triangle math counts risers, not formed treads. For formwork layout it matters: if the porch slab serves as the top tread, you form one fewer tread (6 formed treads for 7 risers in our example).
How do I convert the result to bags?
Divide the waste-adjusted volume by the per-bag yield (80-lb ≈ 0.6 ft³ / 0.017 m³) and round up. Check the bag’s data sheet — yields vary slightly by manufacturer and mix.
Get Your Number
The formula is five operations: derive the riser count by rounding up, compute the actual riser height, take half the tread-by-riser rectangle times the width times the step count, add waste, and convert. The Concrete Stair Calculator runs all five — use this guide to sanity-check its result, especially the riser count and the top-tread question, before you order.
Disclaimer: Calculator results are planning estimates based on the dimensions you enter. Actual concrete requirements vary with site conditions, formwork accuracy, and mix design. Verify volumes with your supplier and consult a licensed professional or your local building authority for structural or code-related decisions.
